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 Circles Unit
Two chords are congruent if and only if (iff) they are equidistant from the center of the circle.
The perpendicular bisector of a chord goes through the center of the circle.
The perpendicular to a chord through the center of the circle bisects the chord.
1. If you go to CK, then you live in NS. TRUE!
The converse to this statement is:
If you live in NS, then you go to CK. NOT TRUE!
You cannot write this as an IFF statement :(
2. Write the converse to:
If a chord is a diameter, then it goes through the center of the circle.
If a chord goes through the center of the circle, then it is a diameter.
A chord is a diameter iff it goes through the center of the circle.
3. Write a converse statement to If a triangle has two congruent angles, then it also has two congruent sides.
If a triangle has two congruent sides, then it has two congruent angles.
A triangle has two congruent sides iff it has two congruent angles.
5 postulates: SSS SAS ASA SAA HL
SSS
If the three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent.
SAS
If the two sides and the included angle of one triangle are congruent to the two sides and included angle of a second triangle, then the triangles are congruent.
ASA
If the two angles and included side of one triangle are congruent to the two angles and included side of a second triangle, then the triangles are congruent.
SAA
If the two angles and non­included side of one triangle are congruent to the two angles and non­included side of a second triangle, then the triangles are congruent.
HL
If the hypotenuse and a leg of one right angle triangle are congruent to the hypotenuse and a leg of a 2nd right angle triangle, then the triangles are congruent
Why isn't SSA helpful for us? Think back to the triangle chapter of grade 11....
Other useful notations will be added to this page soon.....
5 postulates: SSS SAS ASA SAA HL
Ex 1
Given: TP is the perpendicular bisector of AC.
Prove: TAC is an isosceles triangle
Proof:
TP is perpendicular to AC
TPA and TPC are right angles
TPA TPC
AP CP
TP TP
TAP TCP
TA TC
TAC is isosceles
Reason:
Given
given
right angles are congruent
TP bisects AC at P
common side
SAS
CPCTC
definition of isosceles
Ex 2
Given: Figure as marked
Prove: AB=ED
We have looked at triangle proofs to help us prove the circle properties we discovered at the beginning of this chapter.
Prove the segment joining the midpoint of a chord with the center of the circle is perpendicular to the chord.
A circle has a center at the origin and a radius of 13. It has chords AB and CD with points A(­5, 12) B(­12, ­5) C(12,5) and D(­5, 12)
a) prove that AB and CD are congruent
b)prove these chords are equidistant from the center of the circle.